Transmission and reflection of the Rossby waves in a stratified ocean with bottom topography

Transmission and reflection coefficients are calculated for Rossby waves incident on a bottom topography with constant slope in a continuously stratified ocean. The characteristics of the coefficients are interpreted in terms of the quasigeostrophic waves on the slope. In the parameter range where only the barotropic Rossby waves can propagate in the region outside the slope, the bottom trapped wave plays the same role as the topographic Rossby wave in a homogeneous ocean, and hence the transmission is weak unless phase matching takes place. When both of the barotropic and baroclinic Rossby waves can propagate outside the slope, the total transmission can be strong. The bottom trapped wave affects the transmission and reflection, and it leads to the possibility that the Rossby wave is transmitted as a mode different from the incident mode. When the number of the wavy modes on the slope is smaller than that of the Rossby wave modes outside the slope, strong reflection occurs.The results for an ocean with linear distribution of the squared Brunt-Väisälä frequency are compared to those in a uniformly stratified ocean. The weakening of the stratification near the bottom is almost equivalent to reducing the effect of the slope.     

Long nonlinear topographic planetary waves in a rotating stratified ocean

Long nonlinear topographic waves in a continuously stratified ocean with a linear bottom slope are investigated. It is shown that odd cross-channel modes are governed by the Korteweg-de Vries (K-dV) equation. The solitary waves are those of a low pressure type. The long waves are shown to be modulationally stable because of the nonlinear effect due to irrotational motion. All these results are missed if the conventional quasi-geostrophic approximation is adopted.     

Long-period topographic trapped waves in a two-layer ocean with basic flow

The characteristics of free topographic trapped waves are investigated numerically for a two-layer model with basic flow, which is uniform, geostrophically balanced motion flowing parallel to the coast. Six modes are identified for this model with depth variations. They are external and internal Kelvin modes, a topographic Rossby mode, and additional three modes. The two of the additional modes are interesting. The first one is a quasi-geostrophic surface-trapped mode, while the second one is a quasi-geostrophic bottom-trapped mode. It is suggested that baroclinic instability takes place when these two modes take a resonance coupling each other.     

Elementary properties of internal Rossby waves in a two-layer ocean with a uniform current

The elementary properties of internal Rossby waves in a horizontally infinite two-layer ocean with a uniform east-west current, apparently not previously reported in the literature, are documented.     

Stability of planetary waves in a two-layer system

A stability of planetary waves on an infinite beta-plane is investigated in an idealized two-layer fluid system for the large local Rossby numberM. When a primary wave is barotropic, two kinds of barotropic instability modes are found. One of them was previously discussed byGill (1974). When a primary wave is baroclinic, two different kinds of modes that enable barotropic and baroclinic energy transfers are found. The one that has the larger growth rate gains its energy mainly from the mean shear of the primary wave when the internal rotational Froude numberF is smaller than 1/2. WhenF is larger than 1/2, however, the available energy conversion of the primary wave is dominant. This mode has a fairly large part of its energy in the barotropic motion although the primary wave is purely baroclinic.The effect ofO(M ^–2) corrections is found to have a stabilizing influence on all symmetrical modes. The geophysical applications of the present analysis are suggested in the context.     

Dynamics of Rossby waves in an ocean with inhomogeneous currents

A model for a two-layer ocean is applied to consider, in terms of the geometrical optics approximation, the effect of mean flows propagating within the upper layer upon the dynamics of Rossby waves. The case is theoretically analysed, with the depth of the ocean's upper layer much smaller than that of the underlying layer. In this case, the flow's impact upon the baroclinic mode of Rossby waves is ubiquitous, with the exception of synchronicity. Depending on the parameters, four types of wave packets' behaviour in the vicinity of synchronicity points are singled out, namely, the elimination of the peculiarity, shadowing, and convective/absolute instability. For the mean flow profile simulating cyclonic and anticyclonic gyres, we have obtained wave packet trajectories and have studied the wave packet's interaction with the current. Specifically, it has been demonstrated that, given some modulus of the wave packet, vigorous energy exchange between the wave vector and the flow takes place. Translated by Vladimir A. Puchkin.     

The effect of large-scale bottom irregularities on the dynamics of Rossby waves in the ocean

This paper discusses, in terms of the geometrical optics approximation, how large-scale bottom irregularities influence the propagation of Rossby waves in the ocean. To describe the major peculiarities of the phenomenon, a two-layer model is applied, with the depth of the upper layer being considerably smaller than that of the lower layer. However, even with the bottom topography being allowed for, the wave motion is described by two Rossby wave modes, namely, a barotropic mode and a baroclinic mode. It is demonstrated that barotropic mode transformation caused by large irregularities of the sea-floor may lead to wave interaction, resulting in their anomalous distribution. Translated by Vladimir A. Puchkin.     

An investigation on internal solitary waves in a two-layer fluid: Propagation and reflection from steep slopes

Experimental investigations on internal solitary wave (ISW) propagation and their reflection from a smooth uniform slope were conducted in a two-layered fluid system with a free surface. A 12-meter-long wave flume was in use which incorporated with: (1) a movable vertical gate for generating ISW; (2) six ultrasonic probes for measuring the fluctuation of an ISW; and (3) a steep uniform slope (from one of θ=30°, 50°, 60°, 90°, 120° and 130°) much greater than those ever published in the literature. This paper presents the wave profile properties of the ISW recorded in the flume and their nonlinear features in comparison with the existing Korteweg de Vries (KdV) and modified Korteweg-de Vries (MKdV) theories. Experimental results show that the KdV theory is suitable for most small-amplituded ISWs and MKdV theory is appropriate for the reflected ISWs from various uniform slopes. In addition, both the amplitude-based reflection coefficient and reflected energy approach a constant value asymptotically when plotted against the slope and the characteristic length ratio, respectively. The reflected wave amplitudes calculated from experimental data agree well with those reported elsewhere. The optimum reflection coefficient is found within the limit of 0.85 for wave amplitude, among the test runs from steep normal slope of 30° to inverse angle of 130°, and around 0.75 for the reflected wave energy, produced by an ISW on a vertical wall.     

Coastal trapped waves in a two-layer ocean: Wave properties when the density interface intersects a sloping bottom

Properties of coastal trapped waves when the pycnocline intersects a sloping bottom are studied using a two-layer model which has slopes in both layers. In this system there is an infinite discrete sequence of modes, and four different sorts of waves exist: the barotropic Kelvin wave, the upper shelf wave, the lower shelf wave and the internal Kelvin-type wave. They all propagate with the coast to their right in the Northern Hemisphere. The upper and lower shelf waves are due to the topographic-effect on the upper-layer and lower-layer slopes, respectively. Their motions are dominant in the respective layers being accompanied by significant interface elevations. The properties of the upper (lower) shelf wave are almost unaffected by the existence of a lower-layer (upper-layer) slope. The motion of the internal Kelvin-type wave is confined to the region around the line where the density interface intersects the bottom slope.The modes, except that with the fastest phase speed (the barotropic Kelvin wave), are assigned mode numbers in order of descending frequency. Characteristics of Mode 1 change with wavenumber; the upper shelf wave for small wavenumbers and the internal Kelvin-type wave for large wavenumbers (high frequencies). The higher modes of Mode 2 and above can be classified into the upper and lower shelf waves.     

Alternating zonal flows in a two-layer wind-driven ocean

Alternating zonal flows in an idealized wind-driven double-gyre ocean circulation have been investigated using a two-layer shallow-water eddy-permitting numerical model. While the alternating zonal flows are found almost everywhere in the time-mean zonal velocity field, their meridional scales differ from region to region. In the subpolar western boundary region, where the energetic eddy activity induces quasi two-dimensional turbulence, the alternating zonal flows are generated by the inverse energy cascade and its arrest by Rossby waves, and the meridional scale of the flows corresponds well to the Rhines scale. In the eastern part of the basin, where barotropic basin modes are dominant, the zonal structure is formed through the nonlinear effect of the basin modes and is wider than the Rhines scale. Both effects are likely to form zonal structure between the two regions. These results show that Rossby basin modes become an important factor in the formation of alternating zonal flows in a closed basin in addition to the arrest of the inverse energy cascade by Rossby waves. The wind-driven general circulation associated with eddy activities plays an essential role in determining which mechanism of the alternating zonal flows is possible in each region.     

Spurious instabilities of long planetary waves in a two-and-a-half layer model subtropical gyre ocean with a wind-driven steady circulation

In a number of flows that support coupled free-waves, instability results when free-wave dispersion relations calculated without the coupling cross or approach one another. The propagation of long planetary wave perturbations of a two-and-a-half layer model subtropical gyre is one such oceanographically important instance. This note points out that, for a baroclinically unstable two-and-a-half layer model subtropical gyre, numerically aliased long wave dispersion relation plots display extra crossings that are artifacts of the discretization, and these may lead both to spurious numerical instabilities and to numerical misrepresentation of actual instabilities. Paradoxically, the numerical instability may in some instances manifest itself more strongly as the numerical resolution is improved. The aliasing mechanism may be related to the zone of small scale activity found in the southwestern corner of a time dependent model subtropical gyre in the numerical perturbation experiments of (Dewar, W., Huang, R., 2001. Adjustment of the ventilated thermocline. J. Phys. Oceanogr. 13, 293–309). Similar multilayer models are often discussed in the literature, so that the results may be widely useful.     

Topographic Rossby waves in the Gulf of Mexico

Observations of topographic Rossby waves (TRW), using moored current meters, bottom pressure gauges, and Lagrangian RAFOS floats, are investigated for the deep basin of the Gulf of Mexico. Recent extensive measurement programs in many parts of the deep gulf, which were inspired by oil and gas industry explorations into ever deeper water, allow more comprehensive analyses of the propagation and dissipation of these deep planetary waves. The Gulf of Mexico circulation can be divided into two layers with the ∼800-1200 m upper layer being dominated by the Loop Current (LC) pulsations and shedding of large (diameters ∼300-400 km) anticyclonic eddies in the east, and the translation of these LC eddies across the basin to the west. These processes spawn smaller eddies of both signs through instabilities, and interactions with topography and other eddies to produce energetic surface layer flows that have a rich spectrum of orbit periods and diameters. In contrast, current variability below 1000 m often has the characteristics of TRWs, with periods ranging from ∼10-100 days and wavelengths of ∼50-200 km, showing almost depth-independent or slightly bottom intensified currents through the weakly stratified lower water column. These fluctuations are largely uncorrelated with simultaneous upper-layer eddy flows. TRWs must be generated through energy transfer from the upper-layer eddies to the lower layer by potential vorticity adjustments to changing depths of the bottom and the interface between the layers. Therefore, the LC and LC eddies are prime candidates as has been suggested by some model studies. Model simulations have also indicated that deep lower-layer eddies may be generated by the LC and LC eddy shedding processes.In the eastern gulf, the highest observed lower-layer kinetic energy was north of the Campeche Bank under the LC in a region that models have identified as having strong baroclinic instabilities. Part of the 60-day TRW signal propagates towards the Sigsbee Escarpment (a steep slope at the base of the northern continental slope), and the rest into the southern part of the eastern basin. Higher energy is observed along the escarpment between 89°W and 92°W than either under the northern part of the LC or further south in the deep basin, because of radiating TRWs from the western side of the LC. In the northern part of the LC, evidence was found in the observations that 20-30-day TRWs were connected with the upper layer through coherent signals of relative vorticity. The ∼90° phase lead of the lower over the upper-layer relative vorticity was consistent with baroclinic instability. Along the Sigsbee Escarpment, the TRWs are refracted and reflected so that little energy reaches the lower continental slope and a substantial mean flow is generated above the steepest part of the escarpment. RAFOS float tracks show that this mean flow continues along the escarpment to the west and into Mexican waters. This seems to be a principal pathway for deepwater parcels to be transported westward. Away from the slope RAFOS floats tend to oscillate in the same general area as if primarily responding to the deep wave field. Little evidence of westward translating lower-layer eddies was found in both the float tracks and the moored currents. In the western gulf, the highest deep energy levels are much less than in the central gulf, and are found seaward of the base of the slope. Otherwise, the situation is similar with TRWs propagating towards the slope, probably generated by the local upper-layer complex eddy field, being reflected and forcing a southward mean flow along the base of the Mexican slope. Amplitudes of the lower-layer fluctuations decay from the northwest corner towards the south.     

On a decay process of isolated,intense vortices in a two-layer ocean

The dynamics of decaying, isolated, intense vortices are investigated numerically using a highly idealized model. It is found that the cyclonic vortex decays while keeping the ratio TKE/TAPE nearly constant. Meanwhile the anticyclonic vortex decays rapidly within about two years and subsequently decays gradually. This difference is discussed in terms of momentum, vorticity and energy. The limitations of the model are also discussed.     

Response of a simple two-layer ocean to divergent and rotational wind stress

Response of an unbounded two-layer ocean to traveling atmospheric disturbances is analytically investigated. Under the assumption that (1) lower layer is motionless, (2) each layer is homogeneous and in hydrostatic balance, (3) the Coriolis parameter is constant, and (4) the density difference between the two layers is small compared to the density itself, the one-dimensional Klein-Gordon's equation is solved analytically for both the divergent and rotational wind stress. Numerical examples of the wake pattern of the upwelling behind the regional divergent and rotational wind fields are also represented.The results may be summarized as follows: (1) the effect of the divergent wind stress is dominant when the forced Rossby numberR ₀ (the ratio of the frequency of the wind stress to the Coriolis parameter) is larger than unity unless the Mach numberM (the ratio of the traveling velocity of the wind stress to the internal gravitational velocity) is far less than unity, (2) the effect of the rotational wind stress is important, when the forced Rossby numberR ₀ is less than unity.     

Assimilation of simulated altimeter data in a two-layer linear Rossby wave model using the variational method

The variational assimilation method has been examined for ability of reconstructing mesoscale features in altimeter data using a simple dynamic model. A one-dimensional, two-layer Rossby wave model in a cross-track channel has been chosen. The simulated data are constructed from a theoretical solution, which is composed of any combination of two normal vertical (barotropic and baroclinic) modes. The data are collected along tracks and with repeat periods similar to those of the Geosat altimeter. The phase space of control variables is composed of initial and boundary conditions. A cost function is defined to measure differences between the simulated data and the model solution. Regularization (smoothing) terms are also included in the cost function in the form of secon-order spatial and time derivatives of the solution. In this paper, two potential problems existing in the altimeter data assimilation are addressed: one is low cross-track resolution, and the other is vertical projection of the data measured at the sea surface. A succesful metho is developed for reconstructing Rossby waves with wavelengths as short as twice the track intervals for any combination of two vertical modes. A key component to efficient assimilation is a preparation step prior to the actual variational assimilation: a uniform ratio of pressure amplitudes in the two layers is included as an optimization parameter. Starting with the first guess from the preparation step, the variational method is carried out based on adjoint equations without such constraint. Separation of the control variables into the two subsets of the initial and the boundary conditions is found useful. Characteristics of the Hessian matrix are related to the performance of this technique. The method developed for the linear system implies steps to be included in data assimilation for nonlinear meanders and eddies in a major current system as well.     

Generation of internal waves in a three-layer ocean

Within the framework of the linear theory of long waves taking into account the action of the Coriolis force, we solve the problem of generation of internal waves by a barotropic tide impinging on a bottom irregularity of the sea-ridge type. The cross section of the ridge is assumed to be rectangular and the stratification of the ocean is regarded as stepwise with two thermoclines (three-layer model). We study the dependences of the characteristics of generated waves on the parameters of stratification and the period of the impinging barotropic tide. Translated by Peter V. Malyshev and Dmitry V. Malyshev     

Consideration on the topographic control of a baroclinic ocean current

The stability of ocean currents is considered using a two-layer model including the vertical shear in the geostrophic balance and bottom topography. Applying the results, the Kuroshio current along the bottom contour seems to be more stable than any other combination of the current direction and the bottom contour.     

Group velocity and energy transport by Rossby waves

A variational method gives a proof that, for normal modes of Rossby waves in the ocean on a-plane with depth variable in the meridional direction, the time average of the energy flux defined as the product of the pressure and the fluid-particle velocity agrees with the time average of the energy density multiplied by the group velocity. This conclusion holds in considerably general situations and sheds light on the controversial problem of the energy transport by Rossby waves. Moreover, the variational method gives a remarkable relation between the group velocity and the phase velocity. In particular this relation shows that the absolute value of the group velocity never exceeds that of the phase velocity.